Criteria for the origin and quality of pseudospin symmetry in nuclear Dirac mean fields

Pseudospin symmetry (PSS) — the near-degeneracy of single-nucleon levels with quantum numbers $(n-1,l+2,j=l+3/2)$ and $(n,l,j=l+1/2)$ — is a robust feature of nuclear spectra whose physical origin is only partially understood. Chen and Guo (Phys. Rev. C 85, 054312, 2012) scaled the Dirac Hamiltonian so that the relativistic parameter $\lambda=\hbar/mc$ can be continuously reduced to zero, and found that while spin splittings vanish monotonically in the non-relativistic limit, pseudospin splittings increase, stay flat, or decrease depending on the doublet, so PSS cannot be uniquely attributed to relativistic effects. Building on subsequent work (the similarity renormalization group decomposition program, the perturbative interpretation around alternative symmetry limits, and the 2026 first-principles demonstration that PSS emerges at coarse-grained resolution scales), the open problem is to elucidate the origin and quality of pseudospin symmetry in terms of the mean-field potential shapes $\Sigma(r)=V(r)+S(r)$, $\Delta(r)=V(r)-S(r)$, the single-particle quantum numbers, and isospin: to understand which contributions govern the approximate degeneracy and its breaking, and — following the open problem stated by Liang, Meng, and Zhou (2015) — to identify the perturbative pseudospin-symmetry limit and the corresponding symmetry-breaking term in the supersymmetric-quantum-mechanics/SRG framework, beyond the single Woods–Saxon fit to $^{208}$Pb used in the source study and toward a fully quantitative understanding of the origin of pseudospin symmetry and its breaking.

Background

In the nuclear shell model, single-particle levels are organized by the orbital angular momentum $l$ and the spin $s=1/2$, and a strong spin-orbit coupling splits the doublet with $j=l\pm1/2$. When the Dirac equation is used for a nucleon moving in spherically symmetric scalar $S(r)$ and vector $V(r)$ mean fields (the framework of relativistic mean-field, or RMF, theory), it is convenient to form the combinations $\Sigma(r)=V(r)+S(r)$ and $\Delta(r)=V(r)-S(r)$, which govern the upper and lower radial components $G(r)$ and $F(r)$ of the Dirac spinor. It was observed in 1969 (Hecht and Adler; Arima, Harvey, and Shimizu) that single-particle states with quantum numbers $(n,l,j=l+1/2)$ and $(n-1,l+2,j=l+3/2)$ lie surprisingly close in energy; introducing a fictitious pseudo-orbital angular momentum $\tilde{l}=l+1$ and pseudospin $\tilde{s}=1/2$ turns each such pair into a pseudospin doublet with $j=\tilde{l}\pm\tilde{s}$. This pseudospin symmetry (PSS) has been used to explain deformations, superdeformation, identical bands, and magnetic moments in nuclei. Its origin, however, has remained under debate for decades. Ginocchio (1997) showed that $\tilde{l}$ is the orbital angular momentum of the lower component of the Dirac spinor, suggesting a relativistic origin, and Meng et al. (1998) proved that the Dirac equation exhibits exact PSS when $d\Sigma(r)/dr=0$. Subsequent analyses (as reviewed by Liang, Meng, and Zhou, Phys. Rep. 570, 1 (2015)) showed that in realistic nuclei the small pseudospin splitting results from cancellations between several contributions of comparable size — a dynamical character. Typel (2008), using supersymmetric quantum mechanics (SUSY QM, a method relating a Hamiltonian to a partner Hamiltonian with isospectral structure), identified the relativistic harmonic oscillator potential (RHO: $\Sigma$ harmonic-oscillator shaped, $\Delta$ constant) as the simplest exact-PSS limit. Liang et al. (2011) then quantified the perturbative character of the pseudospin breaking relative to each limit: non-perturbative relative to $d\Sigma/dr=0$ (criterion values $|W_{mk}/(E_m-E_k)|\approx0.6$) but perturbative relative to the RHO limit ($\approx0.16$), so the perturbative or non-perturbative character is a property of the chosen symmetry limit as much as of the potential. In parallel, a program based on the similarity renormalization group (SRG, a continuous unitary transformation that diagonalizes a Hamiltonian into a decoupled, band-diagonal form term by term) decomposed the Dirac Hamiltonian into a non-relativistic part, spin-orbit terms, dynamical terms, and relativistic corrections, quantifying each contribution to the (pseudo)spin splittings (Guo, 2012; Li, Chen, and Guo, 2013; Guo et al., 2014, for deformed nuclei). Within this landscape, Chen and Guo (2012) introduced a complementary diagnostic: writing the Dirac Hamiltonian in atomic units with $\hbar=m=1$ turns the Compton wavelength $\lambda=\hbar/mc$ into a tunable parameter, so that the evolution from the relativistic ($\lambda=\lambda_0$) to the non-relativistic ($\lambda\to0$) limit can be followed continuously for Woods–Saxon potentials $\Sigma$ and $\Delta$ fitted to the RMF spectrum of $^{208}$Pb. They found that spin splittings decrease monotonically with decreasing $\lambda$ for every doublet (spin symmetry is purely relativistic), whereas pseudospin splittings increase, stay nearly constant, or decrease depending on the doublet; a perturbative decomposition of the scaled Hamiltonian into operators $O_1,\dots,O_8$ showed that the pseudospin splitting is dominated by the Schrödinger-level part, reduced by the spin-orbit contribution, and enlarged by the dynamical terms, with doublet-dependent rates that produce the non-uniform behavior. Most recently, Ding et al. (2026) showed from first principles that a transition from spin symmetry to pseudospin symmetry occurs naturally as the momentum resolution of chiral and one-boson-exchange nuclear forces is lowered by SRG evolution, providing a microscopic explanation of why PSS appears in low-resolution effective descriptions. What still does not exist is a general, predictive understanding at the mean-field level: criteria, in terms of the shapes of $\Sigma(r)$ and $\Delta(r)$ and the quantum numbers, that tell which mechanism dominates and whether the breaking is perturbative with respect to a stated symmetry limit — including the open problem, stated in the 2015 review, of identifying the perturbative PSS limit and the corresponding symmetry-breaking term — and, ultimately, how large the splitting and its individual contributions are.

Problem Statement

For a nucleon described by the Dirac equation with spherically symmetric scalar $S(r)$ and vector $V(r)$ mean fields, elucidate the origin and quality of pseudospin symmetry in terms of the radial shapes of $\Sigma(r)=V(r)+S(r)$ and $\Delta(r)=V(r)-S(r)$, the single-particle quantum numbers $(\tilde{n},\tilde{l},j)$ — where $\tilde{l}=l+1$ is the pseudo-orbital angular momentum, $\tilde{n}$ the radial quantum number, and $j=\tilde{l}\pm1/2$ the total angular momentum of the pseudospin doublet partners — and the isospin. The goal is to understand and predict, for a given potential and doublet, the following sub-questions, each of which is independently required and independently resolvable: (i) whether the observed near-degeneracy is predominantly produced by the relativistic spin-orbit terms, by the non-relativistic Schrödinger-level structure, or by dynamical cancellations among the competing contributions identified in the perturbative and similarity-renormalization-group decompositions; (ii) whether the pseudospin breaking is perturbative or non-perturbative relative to an explicitly stated exact-PSS symmetry limit; and (iii) the open problem stated by Liang, Meng, and Zhou (2015) of identifying, in the supersymmetric-quantum-mechanics/SRG framework, the perturbative pseudospin-symmetry limit and the corresponding symmetry-breaking term, generalizing the known limit-dependent answer beyond the cases studied to date. A further, optional direction of this program, following the review's call for a fully quantitative understanding of pseudospin symmetry and its breaking, is to predict the sign and the magnitude of the total pseudospin splitting and of its individual contributions (non-relativistic, spin-orbit, dynamical). The goal is posed for spherical mean fields, and the criteria sought must be general in the potential shapes rather than tied to a single fit: they are to be stated for smooth spherical $\Sigma(r)$ and $\Delta(r)$ of arbitrary shape, beyond the single Woods–Saxon parameterization fitted to $^{208}$Pb used in the source study, and illustrated on more than one such potential or nucleus, connecting, where possible, with the first-principles resolution-scale picture in which pseudospin symmetry emerges at coarse-grained scales. Deformed fields, in which the spherical labels $(\tilde{n},\tilde{l},j)$ are replaced by the corresponding deformed single-particle labels, are a natural further test of such criteria rather than part of the required validation.

References

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